An Economic Analysis of Color-Blind Affirmative Action

Roland G. Fryer, Jr. and Glenn C. Loury

with Tolga Yuret∗

Revised: October 2006

Abstract

This paper offers an economic analysis of color-blind alternatives to conventional affirmative

action policies in higher education, focusing on efficiency issues. When the distribution of

applicants’ traits is fixed (i.e., in the short run) color-blindness leads colleges to shift weight

from academic traits that predict performance to social traits that proxy for race. Using data

on matriculates at several selective colleges and universities, we estimate that the short run

efficiency cost of “blind” relative to “sighted” affirmative action is comparable to the cost colleges

would incur were they to ignore standardized test scores when deciding on admissions. We

then build a model of applicant competition with endogenous effort in order to study long run

incentive effects. We show that, compared to the “sighted” alternative, color-blind affirmative

action is inefficient because it flattens the function mapping effort into a probability of admission

in the model’s equilibrium.

∗We are grateful to Ian Ayres, Richard Brooks, Avinash Dixit, Matthew Jackson, Lawrence Katz, Kevin Lang,

Steven Levitt, Adrianna Lleras-Muney, Debraj Ray, and Andrei Shleifer for helpful comments and suggestions. We

also thank seminar participants at numerous universities. Patricia Foo, Laura Kang, Eric Nielsen, and Emily Oster

provided truly exceptional research assistance. Fryer is at the Department of Economics, Harvard University, and

NBER, 1875 Cambridge Street, Cambridge MA 02138 (e-mail: rfryer@fas.harvard.edu); Loury is at the Department

of Economics, Brown University, 64 Waterman Street, Providence RI 02912 (e-mail Glenn_Loury@brown.edu). Fi-

nancial support is gratefully acknowledged from the Andrew W. Mellon Foundation and the Carnegie Corporation.

Fryer thanks the Institute of Quantitative Social Science for generous support.

1

“Implementing race-neutral programs will help educational institutions minimize lit-

igation risks they currently face... If we are persistent in implementing race-neutral

approaches, the end result will be to fulfill the great words of Dr. Martin Luther King

Jr., who dreamed of the day that all children will be judged by the content of their

character and not the color of their skin.”

– U.S. Department of Education: Race-Neutral Alternatives in Postsecondary Educa-

tion: Innovative Approaches to Diversity, Washington, DC., March 2003, pp. 7, 40

1 Introduction

The legal and political climate has shifted dramatically over the last decade on the issue of racial

affirmative action. Accordingly, a number of institutions have begun to reformulate their policies

— particularly in higher education. The states of Texas and Florida now guarantee admission to

their public university systems for all in-state high school students graduating in the top ten and

twenty percent, respectively, of their senior classes.1 In the wake of Proposition 209 — a 1996 ballot

initiative that banned racial affirmative action in California — public higher education officials there

have substantially revised admissions practices.2 Some private institutions have even decided to no

1 In 1996, the state of Texas was ordered by a federal court to eliminate all race-conscious affirmative action

in university admissions decisions. [See Hopwood v. Texas, 78 F.3d 932 (5th Cir. 1996).] The Texas legislature

responded to Hopwood by passing House Bill 588, which guarantees Texas public high school students who graduate

in the top 10% of their class admission to any Texas public college or university.

In February 2000, at the request of Governor Bush, the Florida State Board of Education banned consideration of

race in admissions decisions for the state’s higher education institutions. Florida’s percentage plan, the Talented 20

program, took effect in August 2000. Under this plan, students who graduate from Florida’s public high schools in

the top 20% of their class, complete nineteen specific academic credits, and take an SAT or ACT test are guaranteed

admission to one of eleven state universities, although not necessarily admission to the institution of the student’s

choice.2The so-called “Eligibility in the Local Context” policy was implemented in California in the Fall of 2001. This

program guarantees that the top 4% of each high school graduating class in the state will be admitted to one campus

in the university system. For students admitted in the Fall of 2002, the UC system implemented a “comprehensive

review” policy, which permits each campus to set admissions standards based on ten academic and four non-academic

supplemental criteria, two of which may relate to socioeconomic status.

2

longer require that applicants submit standardized test scores.3 A number of scholars and policy

analysts have urged elite colleges and universities to rely more on the socioeconomic background

and other non-racial, non-academic characteristics of prospective students when assessing their

applications.4

Many justifications can be offered for these changes in admissions practice, but a primary factor

would seem to be the desire to enhance racial diversity amongst the admitted without recourse to

the use of explicit racial preferences. For this reason, we call these types of policies “color-blind

affirmative action,” in contrast to the more conventional, “color-sighted” affirmative action policies.

Under color-sighted affirmative action, selectors give an explicit preference to individual applicants

from some targeted racial group. A commitment to color-blindness prohibits such behavior. Even

so, group-preferential goals can still be pursued tacitly by exploiting knowledge of differences be-

tween the race-conditioned distributions of non-racial traits in the applicant population.5

In this paper, we undertake a theoretical and empirical evaluation of the limits of “race-neutral

approaches” such as those advocated by the U.S. Department of Education in the passage quoted

above. We are particularly concerned with the question of whether, and the extent to which, a

widespread shift toward color-blind affirmative action might be expected to impair the efficiency of

resource allocation in higher education. The answer to this question is of considerable importance

for public policy.6

3Mount Holyoke College, for example, has abrogated that requirement, while committing itself to admit some of

the applicants who do not submit scores.4Richard Kahlenberg (1996) is perhaps the most prominent advocate of so-called class-based affirmative action

policies.5Obviously, introducing a purely random element to the selection process can also raise the yield from any group

that is statistically underrepresented in the pool of admittees. This point is stressed by Chan and Eyster (2003).

However, one important contribution of this paper is to show that the options available to selectors for engaging in

color-blind affirmative action are much broader that the simple use of randomization in the selection process.6 It will be obvious in what follows that the ideas studied in this paper are of quite general relevance. “Color-

blind affirmative action” arises in many areas of public policy having nothing to do with enhancing racial diversity.

For example, a powerful legislator may want to influence the formula specifying how some public benefit will be

distributed among juridisctions, with an eye toward benefiting his own constituency without appearing to be doing

so. More generally, category-blind preferential policies can be used to pursue many group-redistributive goals (among

population segments defined in terms of age, religous belief, gender, health status, region, nationality, and so forth),

when decision-makers wish to avoid the appearance of playing favorites. We elaborate on this point in the Conclusion.

3

There are two distinct ways in which color-blind affirmative action is inherently inefficient. First,

in the short run, when the distribution of traits in the applicant pool may be taken as given, all

affirmative action policies yield lower expected performance among the selected than does Laissez-

faire. This is due to the fact that, under Laissez-faire (i.e., in the absence of any affirmative action

policy), every admitted applicant is anticipated to perform better than any rejected applicant, which

by definition cannot be true under any form of affirmative action. But, color-blind affirmative action

is particularly inefficient in the short run, in the sense that its performance is always dominated by

the best color-sighted affirmative action policy calibrated to achieve the same group representation

goal. This is so because the non-racial factors which best promote selection from a targeted group

are necessarily different from the non-racial factors which best predict post-selection academic

performance — otherwise, some form of affirmative action would not be needed in the first place.7

Secondly, color-blind affirmative action is likely to be inefficient over the longer run as well,

when one considers how the distribution of traits presented by applicants will shift in response

to the incentives created by colleges’ admissions policies. Color-blind policies work by biasing

the weights placed on non-racial traits in the admissions policy function so as to exploit the fact

that some traits are relatively more likely to be found among the members of a preferred racial

group. So, color-blind policies necessarily create a situation where the relative importance of traits

for enhancing an applicant’s prospects of being admitted diverges from the relative significance of

those traits for enhancing an applicant’s post-admissions performance. We show below that this is

never the case under optimal color-sighted policy. Thus, to the extent that color-blind preferential

policies distort applicants’ decisions to acquire performance-enhancing traits prior to entering the

selection competition, additional inefficiencies will emerge.

Our approach is simple and transparent. The central object of our analysis is what we call

the “admissions policy function,” which we imagine to be chosen by a college or university. Given

the applicant pool, this function maps each applicant’s “profile” into a probability of admitting

that applicant. An applicant’s profile is merely a list of that applicant’s “score” along a number

of dimensions, not all of which need be directly related to academic achievement.8 An admissions

7The short run efficiency of color-blind affirmative action depends solely on how well one can proxy for race with

other observable characteristics — and how these characteristics relate to performance.8 In general, an applicant’s chance of admission can be made to depend upon a host of factors. Conventional

academic variables — test performance, grades in high school, recommendation letters, interview results — can be

4

policy function is said to be “color-blind” if, other things being equal, the probability of admission

that gets assigned to a profile does not depend upon an applicant’s race. (Likewise, a policy function

that makes use of race is said to be “color-sighted.”) Since color-blindness is an additional constraint

on the admissions process, given any target rate of admission from a disadvantaged minority group

the best color-blind policy meeting that target must perform less well, from a college’s point of

view, than the best color-sighted policy. Using the College and Beyond Database, we examine data

from matriculates at seven elite colleges and universities to understand the magnitudes involved.

The analysis proceeds as follows:

First, in our simulation exercises we allow colleges to make hypothetical admissions decisions

based on a vector of academic and non-academic applicant traits. This permits us to deduce

how, in the short-run, a broad reliance on color-blind affirmative action might lower selection

efficiency and alter the relative weights given to various factors in the college admissions process

— grades vs. test scores vs. socioeconomic background, for instance. (We wish to stress that our

analytical apparatus is flexible enough to encompass all of the aforementioned color-blind practices

— percentage plans, voluntary test score submission, increased relative weight on non-test score

criteria, preferential admission based on socioeconomic status — as well as conventional affirmative

action policies, within a unified framework.)

Second, in a theoretical model with endogenous applicant effort, we study the ways that color-

blind policies diminish the equilibrium incentives that applicants have to acquire traits valued by

selectors. This gives some sense of the possible longer run efficiency costs of such policies. In the

model, students anticipate the colleges’ policies prior to applying for admission and make a binary,

costly effort decision that affects the distribution of their academic qualifications. In the unique

equilibrium under color-blind affirmative action, as the colleges’ representation target approaches

population parity the fraction of students choosing high effort approaches zero. By contrast, under

color-sighted affirmative action a goal of population parity can be achieved in equilibrium without

vitiating students’ effort incentives. Our principle conclusion is that to rely solely on “race-neutral

approaches” to achieve greater racial diversity in higher education would be to risk some possibly

supplemented with information about an applicant’s social background, life experience, geographic region of origin,

extra-curricular interests, and the like. We think of the specific variables used in an admissions policy function, and

the weights given to them, as being chosen by the college or university in order to meet its admissions objectives.

5

serious, and negative, unintended consequences. (Of course there may be other, non-efficiency-

related, reasons to forego the use of race in college admissions. However, such considerations lie

beyond the scope of this strictly economic analysis.9)

Much has been written on the pros and cons of affirmative action, especially in the labor

market.10 However, until quite recently there had been little attention given in either the theoret-

ical or empirical literatures to resource allocation inefficiencies due to affirmative action in higher

education.11 Two recent contributions warrant to be mentioned. Chan and Eyster (2003) have in-

dependently made one of the observations which we stress here — namely, that a ban on affirmative

action could induce colleges to use inefficient, non-racially preferential means to pursue their racial

diversity ends. They study a constrained-optimal admissions problem for a college that values both

student quality and racial diversity, that can rank students based on a one-dimensional measure of

student ability, but that is enjoined from using racial preferences. They show that in their model

the second-best optimal admissions policy generally involves randomization. Chan and Eyster con-

clude, as do we, that a ban on color-sighted affirmative action could end-up lowering the average

9For a more extended, critical discussion of these race neutral approaches — in the context of a specific legal

dispute over the constitutionality of racial affirmative action at public universities — see the amicus curiae brief filed

with the U.S. Supreme Court in the case Grutter v. Bollinger involving the University of Michigan Law School,

Loury et al. (2003).

We realize, of course, that efficiency is not the only concern when assessing the desirability or the legality of

alternative affirmative action policies. However, under the current Supreme Court’s standards of legal scrutiny, a

racial preference can be permitted if it constitutes a “narrowly tailored” means of furthering a “compelling state

interest.” Thus, once the goal of enhanced racial diversity in college admissions is acknowledged to be a compelling

one, efficiency considerations become relevant to the legal determination of whether a given policy has been narrowly

tailored to advance that purpose. A grossly inefficient policy, relative to some feasible alternative that achieves the

same racial representation goal, is not a narrowly tailored one. See Ayers (1996), and the related discussion in Loury

et al. (2003).10Coate and Loury (1993) develops a theoretical framework for analyzing the incentive effects of affirmative action

in the labor market. Holzer and Neumark (2000) is a comprehensive and insightful review of the theoretical and

empirical literatures on affirmative action.11Datcher Loury and Garman (1993) is an exception. That paper argues empirically that racial preferences in

college admissions may induce an inefficient assignment of minority students to institutions (differentiated by their

degree of selectivity). However, the evidence on this question is mixed. Using different data, Kane (1998) finds no

support for the hypothesis of a detrimental mismatch for minorty students due to (color-sighted) affirmative action

in college admissions.

6

quality of the college’s admitted class. However, their analysis is not comprehensive: It fails to

take into account the fact that colleges can use non-racial proxies, and not just randomization, as

a way to enhance racial diversity under color-blindness. Moreover, they cannot address long-run

efficiency issues at all because, unlike in the present study, their theoretical model treats applicant

characteristics as exogenous.

In another recent paper, Epple, Romano and Sieg (2003) take note of the fact that a prohibition

on explicit affirmative action can be expected to alter a college’s use of non-racial information in

the admissions process. Their numerically simulated model complements ours by focusing on the

supply side of the higher education market. They introduce a framework where colleges differ in

their attractiveness to applicants and compete with one another for the most desirable students.

They are thus able to address the important question (which we here ignore) of how the distribution

of students across a quality-hierarchy of colleges would be affected by a ban on explicit racial

preferences. However they also take the distribution of applicant traits to be exogenous. Overall,

their analysis focuses on a different set of issue than those explored below.

The structure of the paper is as follows. Section 2 describes our empirical model used to

implement color-blind affirmative action and presents estimates of the efficiency losses involved in

the short run. Section 3 develops a model of incentive effects with endogenous traits to illustrate

the long run consequences of the widespread adoption of color-blind affirmative action. Section 4

concludes by discussing how the methods developed in this paper might be applied to other policy

issues. There is a technical appendix which contains all formal results stated in the paper.

2 Color-Blind Affirmative Action in the Short Run

To fix ideas, consider a concrete example of how color-blind affirmative action might work. Sup-

pose initially that a college wants to admit a certain fraction of its applicants while maximizing

the expected performance of those admitted. Let expected performance be a linear function of

standardized test scores and of extracurricular activities in high school.12 It is clear, then, that

this college should adopt the policy of admitting only those applicants whose expected performance

exceeds some threshold, where this threshold has the property that the fraction of applicants exceed-

12We focus in the present example on two varibables likely to enter any college’s admissions policy function without

intending to imply that these are the only variables of interest.

7

A

B

CBAA

LF

Test Scores

Extracurricular Activities

A

B

CBAA

LF

Test Scores

Extracurricular Activities

Figure 1: A Two Attribute Example

ing it just equals the fraction the college desires to admit. In effect, this means that the “weight”

the college gives to activities relative to test scores in its admissions policy function should equal

the ratio of the respective partial correlations of these variables with post-admissions performance.

Now, suppose the college believes that following this threshold policy would lead to “too few”

members of some racial group being admitted.

Imagine that the college wants to obtain a greater degree of racial diversity while continuing

to be race-blind in its treatment of individual applicants. Finally, suppose the college knows that

among its applicants the distributions of activities within racial groups are much more similar to

each other than are the corresponding distributions of test scores. Then the representation of the

racial group with relatively lower (higher) test scores could be enhanced by setting the weight given

to extracurricular activities relative to test scores in the admissions policy function above (below)

the level warranted by the relative correlations of these variables with post-admissions performance.

To introduce such a change in admissions policy would be to engage in the practice of color-blind

affirmative action.

Figure 1 captures the intuition at work. The line segment LF represents a college’s optimal

admissions frontier under a policy of no affirmative action (call this Laissez-faire). Applicants above

the line are admitted with probability one, while those falling below the line are admitted with

probability zero. The line segment CB represents the same college’s admission frontier under a

8

policy of color-blind affirmative action. The CB frontier is steeper than the LF frontier because

we have supposed that extracurricular activities are more nearly equally distributed within racial

groups than are test scores. The shaded area marked A in the figure depicts the set of applicants

(with high test scores and low activities) who are rejected under CB, but who would have been

admitted under LF. The area marked B shows the set of applicants (with high activities and low

test scores) who are admitted under CB, but would have been rejected under Laissez faire. Because

the college intends to fill a fixed number of seats, the number of students falling within each of

these two areas is the same. Yet, because the conditional probability that an applicant belongs to

the targeted racial group is greater given that the application falls in area B than it is given that

the application falls in area A, this college can enhance racial diversity in a race-blind manner by

raising the weight it gives to extracurricular activities relative to test scores when evaluating all

applicants.13

A. An Empirical Model of Short Run Affirmative Action

We now extend and formalize this example. Imagine that a college is to select an incoming

class from a finite set of applicants. Let c denote the proportion of applicants to whom admission

can be offered, 0 < c < 1, and let r denote the target admissions rate for a disadvantaged minority

group (relative to the size of the applicant pool). Let I be the set of all applicants, and take i ∈ I

to index a particular individual.

Suppose that each applicant belongs to one of two racial groups, and let Ri ∈ {1, 2} denote

the racial group membership of applicant i. Each application reports values for a bundle of non-

racial traits (grades, social background factors, test scores and the like). Let J denote the set of

non-racial traits, with specific traits indexed by j ∈ J. Then, the ith student’s application can

be represented by the vector (Ri;xi), where xi ≡ (xji )j∈J , and where xji is the value which the

13Note that this enhanced racial diversity is achieved at the cost of admitting a lower performing class on average,

since the expected performance of every applicant in B is lower than that of any applicant in A.

Furthermore, suppose a college were to make reporting test scores optional for its applicants, while committing

itself to admitting a certain fraction of its incoming class from the set of students electing not to submit scores

(as Mount Holyoke College has, in fact, recently done.) In light of the incentives thereby created for applicants to

selectively report their test scores, this too would be a color-blind policy that, although not explicitly preferential to

any racial group, could be expected to result in more students from low-scoring groups being admitted.

9

application of student i reports for non-racial trait j. Moreover, the college’s entire applicant pool

can be represented by the (large) array X = {(Ri;xi)i∈I}.

Now, in general, an admission policy for the college in this setting associates with every applicant

pool an array of probabilities specifying the chance that each applicant in the pool will be admitted.

Let Ai be the probability of admitting applicant i, 0 ≤ Ai ≤ 1. Then the college’s admissions

problem is to associate with each applicant pool, X, a vector of admission probabilities, A(X) =

(Ai)i∈I , so as to maximize the expected academic performance of the admitted class, subject to

its capacity and racial representation constraints.

Let pi be the college’s expectation of the academic performance of applicant i. We assume that

this expectation can be expressed as a linear function of the applicant’s non-racial traits:

pi ≡ [Expected performance|xi] = β · xi =Xj∈J

βjxji

for some vector of coefficients β. In addition, to the extent that the non-racial traits are distributed

differently within the racial groups, a college could use this fact to predict the racial group mem-

bership of any applicant presenting a particular vector of non-racial traits. Again, adopting a linear

specification of this relationship, we assume that:

ri ≡ Pr[Ri = 2|xi] = γ · xi =Xj∈J

γjxji

for some vector of coefficients, γ.

In what follows, we take it that the vectors of coefficients, β and γ, are known to the college

and enter as parameters in its calculation of an optimal admissions policy. We use our data on

matriculates at several selective institutions to estimate these coefficients. We then use these

estimates to simulate what optimal admissions policies might look like under various regimes at

these colleges, and to evaluate their performance.

We examine the implications of three distinct policy regimes: Laissez-faire (LF); color-sighted

affirmative action (CS); and color-blind affirmative action (CB). We begin by considering the fol-

lowing simple linear program:

max[Ai]i∈I

{(1c)Xi∈I

Aipi}, subject to the following three constraints:

(i) Ai ∈ [0, 1] , i ∈ I, (ii) 1|I|{P

i∈I Ai} ≤ c, (iii) 1|I|{P

i∈I Airi} ≥ r.

10

The maximand above is the anticipated average performance of the admitted class. Constraint (i)

restricts the Ai to being probabilities; (ii) is a capacity constraint; and, (iii) is the affirmative action

representation constraint.

A college’s optimal CB admissions policy must solve this linear programming problem. An

optimal LF policy solves the same problem, but without constraint (iii). An optimal CS policy

can be derived by first partitioning the applicant pool by race, and then solving parallel linear

programs for each group, analogous to the LF version of the program above, but with the group-

specific capacity constraints r2 = r1−λ for group 2, and r1 =

c−rλ for group 1.

Solutions for the optimization problems implied by the three policy regimes are easily derived.

Under the LF regime, one simply orders applicants by their expected performance, admitting the

proportion c with the higher values of pi. That is, for some number, µ (the Lagrangian multiplier

on constraint (ii) above), we have that:

A∗i =

⎧⎨⎩ 1 if β · xi > µ

0 if β · xi < µ.

Here µ must be chosen in such a way that constraint (ii) holds with equality.14

Under the CS regime, there will be separate thresholds for the racial groups. So, for a pair of

numbers µ1 and µ2, with µ1 > µ2, we have:

A∗i =

⎧⎨⎩ 1 if β · xi > µRi

0 if β · xi < µRi

.

Here the µ1 and µ2 are to be chosen such that selection rates for the two groups are consistent with

the capacity and representation constraints holding as equalities.

Under the CB regime, a Lagrangian multiplier on constraint (iii) alters the admissions policy

relative to LF, because non-racial traits are now to be valued both for their association with

prospective academic performance, and for their ability to predict an applicant’s race. Thus, the

optimal CB policy is characterized by two numbers, θ and µ0, such that:

A∗i =

⎧⎨⎩ 1 if [β + θγ] · xi > µ0

0 if [β + θγ] · xi < µ0,

14 If β · xi = µ, then A∗i might need to lie strictly between zero and one for the capacity constraint to hold with

equality.

11

where µ0 and θ are such that both constraints (ii) and (iii) above hold as equalities.

This formalization captures nicely the ideas about color-blind policy mentioned in the intro-

duction. Let j and k be two traits (extracurricular activities and test scores, e.g.). Under LF and

CS regimes, the college’s marginal rate of substitution between traits j and k as reflected in the

admissions policy function, denoted by MRSj,k, is equal to the relative importance of these traits

in forecasting student performance:

MRSj,k =βjβk

,

whereas, under the CB regime, the rate of substitution between traits j and k that holds constant

the probability of being admitted is given by:

MRSj,k =βj + θγjβk + θγk

.

These substitution rates are the signals sent out to applicants about the relative value of various

traits in the admissions process. To the extent that the magnitude (and even the sign!) of such

substitution rates is altered when color-blind means are used to pursue color-conscious ends, the

incentives applicants have to acquire the relevant traits might be badly misaligned. We will now

use our data on student characteristics at selective public and private colleges and universities in

the US to examine how color-blind affirmative action might be expected to play out in practice.

2.1 Simulating the Short Run Impact of Color-Blind Affirmative Action

To apply the foregoing analysis in the context of college admissions in the US we will use actual stu-

dent profiles from the matriculating classes (entering college in 1989) of seven selective institutions

(four liberal arts colleges, labeled “College A” through “College D” and three research universities,

labeled “College E” through “College G” in what follows). We conduct hypothetical admissions

experiments, supposing that the colleges in question would have had to “admit” only a fraction as

many students as were, in fact, admitted.15 Their imagined selection problem is to choose which the

15The ideal data for our thought experiment would come from a double-blind experiment in which neither a

college’s faculty nor its students would know what affirmative action treatment (LF, CS, or CB) they were under.

This would not only solve the obvious selection problems, it would alleviate any externalities that may arise from

faculty or peers knowing that blacks were admitted under a particular regime. Unfortunately, however, these data

do not exist.

12

students to retain, and which not, from among the actual matriculates. The affirmative action goal

is to maintain the original proportion of minority students in this reduced class. We estimate the

loss of efficiency in selection that results from the imposition of the requirement to be color-blind

in the selection process, given this racial representation goal. We also look at the nature of the

constrained-optimal color-blind admissions policies that emerge.

Thus, the capacity constraint for all colleges in these empirical exercises, unless otherwise noted,

is c = 0.5.16 However, the affirmative action representation target under CS and CB varies from

college to college, since the admissions policy maker seeks to maintain the same percentage of

blacks among the selected students as had obtained in the original class, and that percentage varies

across college. Employing the framework just discussed, we model the constrained policy choices in

each regime as linear optimization problems: an admissions policy is chosen, given the distribution

of applicant traits and subject to capacity and representation constraints, so as to maximize the

anticipated average academic quality of the admitted class. Once solutions for these linear programs

are in hand for each college, we can compare the performance of the best admissions policy under

each of the three regimes, and take note of how the constrained optimal color-blind policy attains

its goal through an artful choice of racial proxies.

B. Data Description and Empirical Implementation

The College and Beyond database is remarkably rich — containing student level administrative

data on college performance as well as information on admissions and transcript records of 93,660

full-time students who entered thirty four colleges and universities in the fall of 1951, 1976, and

1989 (see Bowen and Bok 1998 for a complete description). For the purposes of this paper, we

restrict our attention to students from seven institutions in 1989. Our selection criterion is based

solely on the availability of relevant data. Section 5 (the data appendix) describes how we combined

and recoded some of the College and Beyond variables we use in our analysis.

We employ three academic variables: SAT math score, SAT verbal score and High School

Rank; and six socioeconomic background variables: mothers education, fathers education, the

median household income of each student’s zip code, the percent black, Hispanic, and Asian in

each students zipcode, and whether or not each student is related to an alumnus (i.e. a legacy).17

16We will test the robustness of our results to this assumption in the next sections.17Unfortunately, at some colleges family income information was only available for those students who applied for

13

Only one SAT math score and one SAT verbal score was recorded for each student, even if the

student took the test multiple times. Information is not available pertaining to which SAT score

the institution reported. Parental education information was drawn from the student’s college

application. Questions involving parental education varied greatly from university to university.

To account for this, we aggregated the data into two categories: college degree holder or not,

independently, for mother’s and father’s education. Zip Income was calculated by obtaining each

student’s residential Zip Code from College and Beyond dataset and imputing the median household

income by zip code from the 1990 Census. Percent Racial Mix in Zip Code was gleaned similarly.

Legacy status was given if any family member was an alumnus of their university. If legacy status

was unknown, we considered the student a non-legacy.

We use linear regression analysis to associate these variables with the expected class rank after

four years of matriculation achieved by the students in the sample (whose grade histories were

available from the administrative records of the participating institutions). That is, we estimate

models of the form:

GPA_RANKij = Xiβj + εij , (1)

where GPA_RANKij denotes the cumulative GPA of individual i at college j and X captures the

various of academic and social variables described above. We also use these covariates in a linear

probability specification to estimate the conditional likelihood that a given student is of a given

race, given his or her non-racial characteristics, of the following form. 18

Raceij = Xiγj + υij . (2)

To simulate the short run impact of color-bind affirmative action, we estimate equations (1)

and (2) on the full sample of each college, individually.19 With the estimated coefficients, we can

financial aid. So, we have used the median household income in the zip code of residence at the time of application

as a proxy for socioeconomic status.18Note that, while the estimated parameters β and γ are intended to apply to a college’s applicant pool, the data

on which our estimates are based come from matriculates, not applicants, at the various colleges. As a consequence,

the coefficient estimates presented here could be seriously affected by problems of selection bias.19One could also run the regressions within race. We run everything on the full sample so as to not confuse

the inefficiency of race-blindess that occurs because colleges shift emphasis away from academic traits and put more

weight on social traits from the fact that they just have a better regression specification under laissez-faire and race-

conscious admissions. That is, we use a specification on the full sample that is constant under all three regimes to

14

conduct our hypothetical admissions experiments. Under LF, a college will admit a student only ifXj∈J

βjxij ≥ µ. Under CS, a college will admit a student only if

Xj∈J

βRj xij ≥ µR, R ∈ {1, 2} . Under

CB, a college admit only ifXj∈J

£βj + θγj

¤xij ≥ µ0. The latter requires us to solve a simple linear

program to obtain the weights, θ, given to covariates.

C. Selection Problems

The ideal experiment to test the short run inefficiency of alternative affirmative action policies

would randomly distribute such policies across a wide swath of universities. By comparing the

pre and post quality of the admitted classes across similar universities, one can get an unbiased

estimate of the effect of each affirmative action regime. This experiment has not been conducted

and will likely never be.

In lieu of the ideal experiment, we use remarkable data from several elite colleges and univer-

sities. The virtue of our approach is that our empirical model allows one to analyze alternative

affirmative action regimes, holding all else constant, to investigate the effect on ex-post efficiency.

Thus, one can alter the selectiveness of colleges, the covariates observed, or the objectives pursued,

and analyze their effects on the relative efficiency of different affirmative action regimes.

Yet, while the data are rich, any non-experimental analysis has important caveats. Potential

selection problems arise due to (1) lack of data on students who the university did not admit and

their counterfactual performance had they been accepted and (2) the information that admissions

committees use is much richer than the covariates available to researchers. Thus, if we observe

a student admitted with low SAT scores, it is likely that he wrote a stellar essay, had marvelous

recommendation letters, and so on. For our purposes, the former is the most serious threat to

the plausibility of our empirical estimates. Further, our simulations also assume that there are no

spillover effects from attending colleges with higher mean quality. If the quality of a student is a

function of their innate ability and the mean quality of her peers, our estimates will be biased.

The parameter that we are most interested in estimating is the inefficiency that a college can

expect from practicing color-sighted versus color-blind admissions policies, relative to laissez-faire.

The thought experiment is to imagine a college looking at a pool of applicants and comparing

the set of students that they would hypothetically admit under our three policy regimes and the

isolate the desired effect.

15

expected quality of the resulting classes. Yet, because we do not have data on actual applicant pools

(using instead the set of admitted students as the virtual applicant pool and imagining colleges

having to admit half of the students that they, in fact, admitted), the parameter we estimate —

the expected quality of a student admitted in our simulations conditional on being in our sample —

could differ substantially from what we intend. The usual remedies for selection such as estimating

selection equations or re-weighting data are not applicable here because we have no information on

the population of interest.

Whether or not our estimates are reasonable depend on the conditional distribution of academic

traits above and below a college’s selection threshold. If, for example, the conditional distribution

of academic traits is more similar in the applicant pool than our restricted sample and minorities in

this sample have lower academic credentials, our estimates will be biased upward.20 This is likely

the case if the white distribution has a larger right tail among accepted students. Conversely, if

the conditional distribution is less similar in the applicant pool than our restricted sample and the

minorities among the set of applicant who were not admitted are less academically distinguished

than their white peers, our estimates will be biased downward.

We estimated the distributions of predicted college rank for each of our schools, by race, to help

inform which way the selection may go. In all cases, the white conditional distribution (conditional

on being in our sample) dominates the black distribution. Thus, if the conditional distribution

(conditional on not being admitted) of blacks in the applicant pool — which we do not observe —

is similar to whites, our forthcoming estimates of the short run inefficiencies involved in practicing

affirmative action are too large.

In summary, we recognize that selection is a potentially serious problem, but can offer no

compelling remedy.

D. Results

Tables 1 and 2 present summary statistics for our sample of students in four liberal arts colleges

and three research universities, respectively, broken down by institution and racial group. Black

and Hispanic students score over one standard deviation below white and Asian students on the

math and verbal sections of the SAT and have (on average) lower percentile ranks in high school,

20 If black academic credentials are superior to whites, our estimates biased downward.

16

which is consistent with previous research. Among the socioeconomic variables, black students live

in lower income zip codes with a substantial fraction of other blacks and have parents who are less

likely to be college educated and dramatically less likely to be an alumnus of their child’s college.

A similar pattern holds for Hispanics — yet Asians outperform whites on the math section of the

SAT, live in higher median income zipcodes, and are more segregated from Blacks and Hispanics.

Table 3 reports results from the college-specific regression equations, which used academic and

socioeconomic background variables to predict a student’s class rank after four years of matricu-

lation (equation 1). The interpretation of the coefficients is standard: a one unit change in the

independent variable, all else constant, produces the reported change in the dependent variable.

For instance, the coefficient on SAT math in College A is 4.04. This means that a 100 point increase

in SAT scores is associated with a 4 point higher GPA rank at college graduation.

Interestingly, both a student’s SAT verbal score and their high school rank are stronger predic-

tors of college rank upon graduation than is his or her SAT math score. Parental education is also

a strong predictor of college rank. After controlling for our three academic variables and parental

education, the average income of a student’s zip code is not statistically significant. Zipcode racial

demographics, however, are important predictors of college performance, even after controlling for

other pre-college characteristics. In six out of seven colleges, a student’s college performance is pos-

itively (and significant) related to the fraction of Asians in that student’s zipcode, and negatively

related to the fraction of Hispanics. All else equal, the fraction of blacks in a zip code are negatively

related to college performance for all seven colleges. It is likely that these racial demographic corre-

lations are capturing unobserved school or neighborhood quality factors that promote performance

in college, but that are not captured in standardized tests. Perhaps the most surprising finding of

Table 3 is that more than 60% of student variation in college rank remains unexplained at all of

the colleges, after taking account of students’ precollege characteristics.

Table 4 reports results from the auxiliary regressions that we imagine the colleges to have

run if, when operating under a color-blindness constraint, they needed to use academic and social

background variables to forecast the likelihood that a student is black. As intuition might suggests,

Blacks are more likely to live in own race neighborhoods, reside in lower income zipcodes, be

non-legacies, and have lower scholastic achievement.

Tables 5 through 7 report the results of greatest interest, regarding the relative inefficiency

17

of race neutral alternatives (Table 5), the implication of such policies for the representation of

various racial groups (Table 6), and the way that optimal color-blind affirmative action alters the

weight given to various factors in the optimal admissions formula — i.e., test scores, grades and

socioeconomic background measures (Table 7).21 Bear in mind that we measure the performance

of a policy in terms of the average of the class rank predicted for the students admitted under that

policy.

Table 5 reports our estimates of the relative performance of conventional (i.e., color-sighted)

affirmative action, and four alternatives policies (Laissez-faire performance at each college has been

normalized at 100.) The columns in the table represent institutions of higher education. The first

four (A-D) are liberal arts colleges, and the last three (E-G) are research universities. The rows

represent five different policy regimes: random admissions; LF policies when colleges are artificially

prohibited from knowing students’ SAT scores and high school grades, respectively; color-sighted

affirmative action; and color-blind affirmative action. As theory predicts, in every case color-

blind policies perform less well than do color-sighted policies. The magnitude of efficiency loss

from employing color-blind rather than color-sighted affirmative action varies across institutions,

ranging from less than one percentage point at College B to more than 6.6 percentage points at

College D (a small, elite institution in the northern portion of the US). Notice that at most colleges,

the loss of selection efficiency associated with going from color-sightedness to color-blindness (given

the same representation target) is comparable to, and sometimes even exceeds, the loss of efficiency

that would arise if colleges had no interest in racial representation, but were constrained from using

students’ grades or test scores in the admissions process. (That random admissions costs no college

more than 20% in efficiency is clearly due to the fact that our sample consists of matriculates,

not applicants: grossly “underqualified” individuals are unlikely to have been admitted to these

selective institutions and, therefore, are significantly underrepresented in our sample.)

Table 6 shows the consequence of our three policy regimes for the overall ethnic/racial com-

position of the admitted class. Because color-blind affirmative action shift weight from academic

characteristics to social characteristics, such policies directed toward blacks will concurrently help

21The LF and CS optimal admissions policies both use the same weights (those derived from the regression

predicting college class rank), while the CB policy employs weights that are “biased” in order to exploit the fact that

some variables are more closely correlated (positively or negatively) than are others with a student’s being black.

18

Hispanics and low-income whites, whereas color-sighted affirmative action will not. Table 6 also

sheds light on the inefficiencies described above. Consider College D. Under a LF regime, the col-

lege admits no black students. This implies that all blacks are in the left tail of the academic trait

distribution, conditional on being accepted. But, affirmative action requires college D to admit

16 black students which explains the efficiency hit. Conversely, College B would admit 5 black

students on their own accord and are required to admit only 7 more due to affirmative action. As

such, College B has very little loss in efficiency in the short run.

Table 7 reports our calculations of the weights on students’ characteristics in the admission

formula that are employed under optimal LF and CB policies. In effect, colleges are assigning

a score to each student, and admitting that half of the applicant pool with higher scores. The

numbers in Table 5 are simply the coefficients used in a linear formula to derive a student’s score

from that student’s academic and socioeconomic traits. It is clear from the illustrative empirical

results reported in Table 7 that optimal CB admissions policy gives less weight to test scores, more

weight to high school grades, and more weight to social background factors than does optimal

policy under the LF-CS regimes. As such, it is not surprising that CB admissions policy targeted

on blacks tends also to raise admissions rates for Hispanics while lowering them for whites and

Asians, as Table 6 reveals. This is the crucial point in which these estimates and the theoretical

model developed in the previous section interact: The short run efficiency of color-blind affirmative

action may not seem so great (and it would shrink toward zero if we could add more variables that

are correlated with race.) Yet, because color-blind policy shifts weight from performance-related

traits to social characteristics that are weakly correlated with achievement, it lowers incentives for

applicants to invest in the traits valued by selectors.

Throughout the above analysis, we made two assumptions that warrant further emphasis. First,

we imagined that the colleges could admit only half of their original matriculating classes. To gauge

the importance of that assumption, Figure 2 plots the relative efficiency of various affirmative action

policies at each college while allowing colleges to admit a proportion of their original classes that

ranges from 10% to 90%. One can see from the figure that the tighter is the capacity constraint (i.e.,

the lower the “percent admitted”), the more efficient is CS policy relative to random admissions.

Yet, the inefficiency of CB relative to CS policy is essentially independent of the percent admitted

at all institutions except College D. We conclude that our results are not sensitive to this first

19

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

10 20 30 40 50 60 70 80 90

total percent admitted

ratio

of r

elat

ive

effic

ienc

ies:

C

B/C

S

FAEBGCD

0.6

0.7

0.8

0.9

1

10 20 30 40 50 60 70 80 90

total percent admitted

rela

tive

effic

ienc

y: R

ando

m/C

S

FAEBGCD

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

10 20 30 40 50 60 70 80 90

total percent admitted

ratio

of r

elat

ive

effic

ienc

ies:

C

B/C

S

FAEBGCD

0.6

0.7

0.8

0.9

1

10 20 30 40 50 60 70 80 90

total percent admitted

rela

tive

effic

ienc

y: R

ando

m/C

S

FAEBGCD

Figure 2: Ratios of Relative Efficiencies

assumption.

Second, we assumed that colleges strive to achieve the same level of diversity under our hypo-

thetical color-blind constraints as they had actually achieved under what we must presume to have

been a color-sighted system. But, this assumption is implausible: if imposing blindness raises the

cost of affirmative action, it stand to reasons that colleges would then consume less of it! Further,

as Sander (1997) demonstrates, the marginal cost to academic goals of racial affirmative action can

rise rapidly with the percent minority students being admitted. Given this finding, it is plausible

that the relative efficiency of color-blind affirmative action might also vary with the magnitude

of the racial representation target. Figure 3 tries to detect such non-linearities in our data. The

x-axis measures the affirmative action goal of each school. The y-axis measures the efficiency of

20

0.8

0.85

0.9

0.95

1

0 0.01 0.02 0.03 0.04 0.05

Affirmative Action Goal

Rel

ativ

e C

olor

Blin

d Pe

rfor

man

ce

College ACollege BCollege CCollege DCollege ECollege FCollege G

Figure 3: The Tradeoff Between Efficiency and Representation

color-blind affirmative action relative to laissez-faire. Interestingly, as Sander (1997) observed in a

different context, we find a distinct non-linearity involved in most schools between representation

and efficiency. (Notice that College D, where color-blindess was most inefficient, also has the most

stark non-linear relationship.)

3 The Long Run Consequences of Color-Blind Affirmative Action

There are several reasons to expect that color-blind policies may undermine the efficiency of the

selection process in the long run. Unfortunately, with the current data it is not possible for us to

empirically estimate the potential magnitude of these long-run effects.22 However, by formulating

a rigorous theoretical model of the selection problem with endogenous traits, we can gain some

22One could try to estimate the impact of the percentage plans discussed in the introduction on student effort,

using data across states over time. Given a measure of effort (average SAT scores or school attendance rates, e.g.)

in various states over several years, a difference-in-differences model could be estimated by contrasting student effort

before and after the implementation of affirmative action, as between states using blind versus those using sighted

policies. Yet, since the adoption of blind rather than sighted policies are clearly not exogenous at the state level, it

is not clear what would be learned by such an exercise.

21

insight into the main issues.23

Imagine, then, that a continuum of applicants (students) of unit mass consists of two racial

groups, R ∈ {1, 2}, where λ ∈ (0, 1) is the proportion belonging to group 1. Applicants seek to be

accepted by any one of a large but finite number, N , of identical firms (colleges). Each applicant is

randomly assigned to a firm, and so each firm faces an applicant pool of measure 1/N that is the

statistical replica of the overall population. Let ω > 0 be the gross value to an applicant of being

accepted. Each firm can accept at most the fraction c ∈ (0, 1) of those who apply. Firms prefer to

accept the better-qualified applicants, and take the distribution of characteristics in their applicant

pools as given, independent of their acceptance policies.

Prior to being assigned, applicants make an ex ante binary effort decision e ∈ {0, 1} that affects

their qualifications ex post. The incentive effects of affirmative action will be reflected in this model

by the way that alternative policies alter the distribution in the student population of this binary

effort variable, and the resulting distribution of qualifications. We assume that low effort (e = 0) is

costless, but high effort (e = 1) entails a cost, k ≥ 0, for an applicant. The frequency distribution

of effort cost differs between racial groups. Let GR(k) be the fraction of group R with effort cost

less than or equal to k , and let gR(k) be the associated density function. Let G(k) = λG1(k)+

(1 − λ)G2(k) be the CDF of effort cost for the overall population, with g(k) being the associated

population density function. We assume group 2 is “disadvantaged” in the sense that it has a

uniformly less favorable cost distribution than group 1: [g1(c)/g2(c)] is a monotonically decreasing

function of c.

An applicant’s qualification (as perceived and valued by firms) is a stochastic function of effort.24

Let t be a number representing an applicant’s qualification, let Fe (t) be the probability that effort

e leads to a level of qualification less than or equal to t, and let fe(t) be the associated density

23Our model implicitly assumes (1) that the law permits the pursuit of racial diversity with policies that are not

explicitly contingent on an applicant’s race; and, (2) that perfect proxies for race do not exist (i.e., there are no

variables which, taken together, are perfectly correlated with race while being unrelated to college performance).24We emphasize that no asymmetry of information between firms and applicants is being assumed here. Qualifi-

cations are perfectly and costlessly observable by firms. Our assumption is that, when an applicant chooses effort ex

ante, the extent of qualification that results ex post is random at the individual level. Because there is a continuum

of applicants, the distribution of qualifications in any population depends only on the fraction of applicants in that

population who have chosen high effort. We assume that firms care only about an applicant’s qualifications ex post,

and not about the effort taken ex ante.

22

function. High effort is assumed to increase an applicant’s qualification in the following sense:

[f1(t)/f0(t)] is a monotonically increasing function of t.

Finally, let πR represent the fraction of applicants in group R who choose action e = 1 (R = 1, 2),

with π being the fraction of all applicants who exert high effort. The variables πR are endogenous,

and will depend on incentives for applicants to take high effort created by the firms’ acceptance

policies. In a population where the fraction π exert high effort, the CDF of the distribution of

qualifications is denoted F (π, t); f(π, t) denotes the associated density function.

An applicant is characterized by his or her racial group and degree of qualification. A firm’s

acceptance policy must be some function A(R, t), representing the probability that an applicant of

racial group R and qualification t is accepted. A firm’s policy is color-blind if A(1, t) = A(2, t) for

almost every t.

We consider the behavior of firms under three possible policy regimes: Laissez-faire, color-

sighted and color-blind affirmative action (LF, CS, and CB, respectively). In each case, we assume

firms take the proportion of high effort applicants in each racial group as given when deciding upon

an acceptance policy.

Under LF, firms are unconcerned with diversity so they ignore group identity information. Thus,

given the proportion of high effort applicants, firms choose an acceptance policy which maximizes

the expected quality of those admitted, subject to their capacity constraint. The best LF policy

is a color-blind threshold policy, where firms accept the fraction c of the applicant pool with the

highest qualifications.

Let r∗2 be the acceptance rate for group 2 that obtains under this LF optimal policy. We

formalize affirmative action (either the color-blind or the color-sighted variety) by positing that

firms seek an acceptance rate for group 2 members, r2, that exceeds r∗2, but is no greater than that

implied by population parity. Given their beliefs about the fraction in each group of applicants

who have chosen high effort, we require firms to choose an acceptance policy under which they

anticipate to accept group 2 applicants at the rate, r2. The aggressiveness of the affirmative action

policy pursued by firms is taken to be exogenous throughout this analysis. In light of the capacity

constraint, and given this two-group set-up, a representation target for group 2, r2, necessarily

implies a target for group 1, r1.

Consider now a firm’s selection problem under a CS policy regime with representation target

23

r2. Taking the fraction of high effort applicants in each group as given, firms choose an acceptance

policy for each group which maximizes the expected quality of those admitted, subject to their

capacity constraint and ensuring that group 2 applicants are adequately represented among those

who are selected. Under color-sighted affirmative action firms follow distinct threshold policies for

each group, accepting the fraction rR of the group R applicant pool with the highest qualifications,

R = 1, 2.

Lastly, consider the firms’ behavior in the CB regime, with the representation target for group 2

given. Firms again take the distribution of high effort in both groups as given, but now must choose

a color-blind acceptance policy function so as to maximize mean qualifications of those accepted,

while anticipating to generate the desired representation of group 2 members. The key issue is that

a firm’s problem cannot be a function of racial identity. This problem is a linear program in an

infinite dimensional space. Such programs are studied extensively in Anderson and Nash (1987),

though our technical appendix shows that we need not solve this problem explicitly to characterize

the equilibrium distribution of applicant qualifications under a CB policy regime in this model.

Imposing the color-blind constraint on firms that remain intent on achieving more representation

for the disadvantaged group than occurs under Laissez-faire must lead to a situation in which some

applicants are accepted while others with higher qualifications are rejected, thereby undercutting

applicants’ incentives to exert high effort. This is a general feature of color-blind affirmative action

policies, and it is the basic reason that such policies must be inefficient over the longer run, relative

to the color-sighted alternative.25

25This property would not hold if, in the manner of Chan and Eyster (2003), we were to impose some kind

of monotonicity constraint on firms [e.g., requiring that A(t) be non-decreasing, out of the incentive compatibility

concern that applicants not see any gain from under-reporting their qualifications.] Still, the basic point we are

making here would remain valid, even were we to impose monotonicity. Under such a constraint, the firm’s problem

can be reformulated so that it becomes (the dual of) what Anderson and Nash (1987, section 4.4) call a “continuous

semi-infinite linear program.” If we apply their Theorem 4.8 (page 76) to this reformulated problem, we can conclude

that with a monotonicity constraint the firm’s optimal acceptance policy can be expressed as a step function with at

most two points of discontinuity. This, in turn, implies that there will be levels of qualification t and s, with t < s,

such that A∗(t) > 0 and A∗(s) < 1. That is, some applicants are accepted with a probability strictly greater than

zero, while others with higher qualifications are accepted with a probability strictly less than one, again undercutting

applicants’ incentives to exert high effort.

But, the main point we wish to emphasize is that, once applicants’ qualifications are allowed to be endogenous

24

Regardless of the regulatory regime, an applicant with exogenous characteristics (R, k) who

anticipates firms to employ the acceptance policy A(R, t) will exert high effort only if the costs of

doing so are no greater than the benefit.

An equilibrium in this model is an acceptance policy for firms and an effort supply function for

applicants, that are mutual best responses to one another (see Proposition 1, Appendix).

The Impact of Affirmative Action on Applicant Qualifications in Equilibrium

To describe how affirmative action policies affect the equilibrium distribution of qualifications

among applicants in the two groups in our model we must introduce some additional terminology.

If in LF equilibrium the marginal applicant has low (high) qualifications, then we will say that

acceptance standards are “loose” (“tight”.) Furthermore, if the representation target r2 ≈ r∗2(c)

(r2 ≈ c) then we will say that the affirmative action goal is “weak” (“strong”.)

Proposition 2 in the appendix establishes the following set of results. (1) If standards are

loose in LF equilibrium, then the pursuit of sufficiently weak affirmative action goals with CS

policies increases qualifications among the advantaged group and decreases qualifications among

the disadvantaged group, thereby widening the racial qualifications gap. (2) If standards are tight

in LF equilibrium then weak CS affirmative action decreases qualifications among the advantaged,

and increases qualifications among the disadvantaged, thereby narrowing the racial qualifications

gap. And, (3) if Laissez—faire equilibrium standards are neither loose nor tight, then sufficiently

strong CS affirmative action goals must decrease the qualifications of both groups.

Finally, Proposition 3 considers the effect of color-blind affirmative action on applicant qualifi-

cations in equilibrium. To state the result we need one last definition: We will say that a selection

problem is characterized by “elitism” if no feasible acceptance policy by firms can induce “high

cost” applicants to exert high effort. Now, suppose the condition of elitism obtains. Then we show

in the Appendix that: (1) for every affirmative action goal r2, there is a unique color-blind af-

firmative action equilibrium; (2) as r2 rises, the level of qualifications in both groups declines, as

does the qualifications gap between the groups; (3) in the limit, as r2 approaches c (population

in the manner that we follow here, the imposition of such a monotonicity constraint on the firm’s acceptance policy

is irrelevant for determining the distribution of qualifications in equilibrium. This is because (as we show in the

Appendix,) given the capacity and representation constraints, all feasible color-blind affirmative action policies for

firms generate the same (diminished) effort incentives for applicants.

25

parity as a goal), the proportion of applicants choosing high effort in the color-blind affirmative

action equilibrium approaches zero; however, (4) there does not exist an equilibrium in pure strate-

gies under color-blind affirmative action that implements the representation target of population

proportionality, r2 = c.

This last is a stark result that warrants further emphasis.26 Consider an extreme example: one

way to achieve population proportionality for all groups is to select from among candidates for a

limited number of positions at random, with every applicant facing the same chance of success. This

assures (with large numbers of applicants and statistical independence of applicant traits) that the

fraction of successful candidates from any group equals the fraction of applicants from that group.

Yet, random selection gives applicants no incentive to acquire traits valued by the selector. In

equilibrium, the population of applicants (from all groups) will be much less distinguished under

random selection, despite the fact that those selected will indeed be racially diverse. Appendix

Proposition 3 demonstrates that the intuition of this extreme example extends to the general

case.27

Our principle theoretical conclusion is that color-blind affirmative action entails a basic trade-

off between incentives and representational goals. If firms are constrained to be color-blind but

continue to value diversity, they will act in such a way as to “flatten” the function that relates

a worker’s probability of being accepted (in equilibrium) to that worker’s level of qualification:

Some lower qualification workers must have a greater chance of being accepted under color-blind

affirmative action, and some higher qualification workers must have a smaller chance. (Otherwise,

the disadvantaged group, which has relatively more low qualification members, cannot have its

representation increased.) This flattening of the link between qualifications and success undercuts

incentives for all workers to exert preparatory effort by reducing the net benefit of investment.

Beyond the narrow definitions of efficiency employed in this paper, there are at least two sce-

26We have further results along these lines in a more general setting. See Fryer and Loury (2003), which con-

siders the problem of equilibrium and optimal handicapping (i.e., affirmative action or, more generally, “categorical

redistribution”) in winner-take-all markets.27There is an interesting externality here that promotes long run inefficiencies. Individual selectors drawing on a

large, common pool of prospective applicants, may not take into account that their choice of selection criteria alters

the distribution of traits in the overall applicant population. This makes the adoption of a random selection method

look like a low cost move for any given selector. But when all selectors make this choice, they are all worse-off than

they would be if none of them made it.

26

narios where color-blind affirmative action might be preferred to color-sighted policies. The most

obvious is that color-blind approaches may be more viable politically. Diversity-promoting policies

not explicitly contingent on an applicant’s race seem, in the current climate, either to fly below

the legal radar screen, or simply to be less objectionable than explicitly racial policies.28 Secondly,

to the extent that one is concerned with the relative reputation of minorities within firms, color-

blindness may be preferred since, although a “blind” selection mechanism reduces the reputation

of the average selected individual relative to a “sighted” policy, it increases the relative reputation

within firms of selected members from the preferred group.

4 Beyond Race

The issues explored in this paper are of more general interest, beyond the study of racial equity.29

There are many context in which a firm or public authority distributes some resource across a

heterogeneous, categorically diverse population, with the dual objectives of allocating that resource

to the most productive members of the population while avoiding an undue categorical disparity in

receipt of the benefit.30 A state government may need to distribute funds for public works among

competing cities and towns, aiming to allocate the funds where they are most needed (or can best

be made use of), while limiting any resulting disparity amongst jurisdictions. Similarly, a supplier

of consumer credit (or insurance) may need to screen applicants according to creditworthiness (or

insurability), without thereby generating a customer base with “too few” racial minorities. When

the observable individual traits that are positively associated with creditworthiness (or insurability)

are less frequently present in one population group than another, then simply screening-out the

least qualified applicants could lead to a stark disparity in rates of selection between groups. For

political, economic or legal reasons, such an outcome might be undesirable. However, it may also

be undesirable in such settings to explicitly discriminate among applicants based on (race or sex-

defined) group identities. This situation leads to the posing of an analytical problem nearly identical

28Justice O’Connor in Croson (Richmond v. J. A. Croson Co., 488 U.S. 469) and Adarandand (Adarand Con-

structors v. Pena, 515 U.S. 200) can be read as affirming the latter view.29There is a clear relation between our analysis of color-blind affirmative action and policy targeting in international

trade. See Bhagwati (1971) for a nice survey. We are grateful to Avinash Dixit for pointing us to this literature.30Akerlof (1978) investigates a related problem of “tagging” in the context of optimal taxation and welfare pro-

grams.

27

to the one investigated in this paper.

Alternatively, consider a customs union — e.g., the European Common Market. Imagine that

a member state wants to favor its domestic producers of some good, but cannot do so directly

without violating the trade agreement. Imagine further that members of the customs union are

permitted to impose quality standards regulations which all goods, no matter where they originate,

must meet. For instance, some Germans may want to limit imports into their country of Dutch

beer, but may be forbidden to bar such products by Common Market rules. Still, they can require

of any beer sold in Germany that it have so much hops, so little preservative, come in kegs that

are made of a particular wood, etc.31

More generally, let the country have some preferences about what its quality standards should

be, and suppose that the relative costs to domestic and foreign producers of meeting different

standards are known. Suppose quality has two dimensions and that, compared to foreign producers,

domestic firms are at an absolute cost disadvantage when forced to produce the Laissez-faire optimal

quality vector. So, domestic producers would get a relatively low market share under the Laissez-

faire optimal (i.e., disinterested) quality regulations. However, suppose domestic firms have a

comparative cost advantage over foreigners in satisfying one dimension of quality. Then, by biasing

regulation so as to give greater importance to that dimension of quality, the country in question

can raise its domestic firms’ market share without appearing to practice protectionism, but at the

expense of having a less than optimal (given their natural preferences for quality) set of regulations.

Again, we have arrived at a formulation analogous to the model studied here.

References

[1] Akerlof, George. 1978. “The Economics of ‘Tagging’ as Applied to the Optimal Income Tax,

Welfare Programs, and Manpower Planning,” American-Economic-Review, 68(1): 8-19

31 Indeed, there is a real case involving Beck (German) and Heineken (Dutch) beers, in which the European Court

of Justice prohibited Germany from enforcing its purity requirements for beer against beverages imported from other

members of the EC. (See Commission of the European Communities v Federal Republic of Germany, Case 178/84,

Judgment of 12 March 1987, 1987 ECR 1227.) We are grateful to William James Adams for bringing this example

to our attention.

28

[2] Anderson, E.J. and P. Nash. 1987. Linear Programming in Infinite Dimensional Spaces: Theory

and Applications. New York, NY: John Wiley and Sons.

[3] Ayers, Ian. 1996. “Narrow Tailoring,” U.C.L.A. Law Review, 43:6, 1781.

[4] Bhagwati, Jagdish. 1971. “The Generalized Theory of Distortions and Welfare,” in Bhagwati

et. al. Trade, Balance of Payments and Growth, North Holland (Kindelberger Festschrift).

[5] Bowen, William and Bok, Derek. 1998. The Shape of the River: Long Term Consequences of

Considering Race in College and University Admissions. Princeton, NJ: Princeton University

Press.

[6] Chan, Jimmy and Eyster, Erik. 2003. “Does Banning Affirmative Action Lower College Student

Quality?” American Economic Review, 93:2 (June), 858-872.

[7] Coate, Stephen. and Glenn C. Loury. 1993. “Will Affirmative Action Eliminate Negative

Stereotypes?” American Economic Review, 83 (5), 1220-1240.

[8] Datcher Loury, Linda and David Garman. 1993. “Affirmative Action in Higher Education.”

American Economic Review, 83 (2), 99-103.

[9] Epple, Dennis, Richard Romano and Holger Sieg. 2003, “The Practice and Proscription of

Affirmative Action in Higher Education: An Equilibrium Analysis,” NBER Working Paper

#9799, June 2003

[10] Fryer, Roland and Loury, Glenn. 2003. “Categorical Redistribution in Winner-Take-All Mar-

kets.” mimeo. Harvard University.

[11] Holzer, Harry and David Neumark. 2000. “Assessing Affirmative Action,” Journal of Economic

Literature, 38:3 (September), 483-568

[12] Kahlenberg, Richard D. 1996. The Remedy: Class, Race and Affirmative Action. New York:

Basic Books.

[13] Kane, Thomas J. 1998. “Racial & Ethnic Preferences in College Admissions,” chp. 12 in C.

Jencks and M. Phillips (eds.), The Black-White Test Score Gap. Washington D.C., Brookings

Institution Press, 431-456.

29

[14] Loury, Glenn C., et al. 2003. “Brief of Social Scientists as Amici Curiae in Support of Re-

spondents,” in the Supreme Court of the United States, Barbara Grutter, Petitioner, vs. Lee

Bollinger et al., Respondents, No.02-241 (www.bu.edu/irsd/MichiganBrief.doc).

[15] Sander, Richard. 1997. “Experimenting With Class-Based Affirmative Action,” 47 Journal of

Legal Education 472-503.

30

5 Appendix: Technical Proofs

In this section, we provide a formal derivation of the model and results in section 3.

A. Firm Behavior

An applicant is characterized by the pair (R, t). A firm’s acceptance policy must be some func-

tion A(R, t), representing the probability that an applicant with characteristics (R, t) is accepted.

Definition 1 A firm’s policy is color-blind if A(1, t) = A(2, t) for almost every t.

Under LF: Firms choose the function {A(t) : 0 ≤ t ≤ 1}, with 0 ≤ A(t) ≤ 1, so as to:

max{Z

tA(t)f(π, t)dt}, subject toZ

A(t)f(π, t)dt = c.

The solution to this problem is given by: {A∗(t) = 1, t ≥ t∗and A∗(t) = 0, t < t∗}, where

F (π, t∗) = 1− c.

Consider now a firm’s selection problem under a color-sighted affirmative action policy regime,

with representation target r2. Taking (π1, π2) as given, firms choose {A(R, t) : 0 ≤ t ≤ 1}, with

0 ≤ A(R, t) ≤ 1, so as to:

max{Z

tA(R, t)f(πR, t)dt}, subject toZ

A(R, t)f(πR, t)dt = rR, R ∈ {1, 2}.

Here the problem is solved separately for each group. As before, it is clear that the solution to

this problem involves {A∗(R, t) = 1, t ≥ t∗R and A∗(R, t) = 0, t < t∗R}, where F (πR, t∗R) = 1− rR,

R = 1, 2. Thus, under color-sighted affirmative action firms follow distinct threshold policies for

each group, accepting the fraction rR of the group R applicant pool with the highest qualifications,

R = 1, 2.

Consider the firms’ behavior in the CB regime, with the representation target for group 2 given

as r2 < c. Firms again take (π1, π2) as given, but now must choose a color-blind acceptance policy

function so as to maximize mean qualifications of those accepted, while anticipating to generate

the desired representation of group 2 members. So, the problem for firms becomes choosing A(t)

so as to:

max{Z

tA(t)f(π, t)dt}, subject toZ

A(t)f(πR, t)dt = rR, R = 1, 2

31

An equivalent way of expressing the firm’s problem under a CB regime is as follows:

max{Z

tA(t)f(π, t)dt}, subject toZ[A(t)f(π, t)dt] = c and

Z[A(t)f(π, t)ξ(π1, π2, t)dt] = (1−λ)r2,

where

ξ(π1, π2, t) =(1− λ)f(π2, t)

f(π, t).

Thus, ξ(π1, π2, t) is the conditional probability that an applicant belongs to the group 2, given

group specific high effort rates (π1, π2), and given that the applicant’s level of qualification is

t. This problem is a linear program in an infinite dimensional space, as both the objective and

the constraints may be regarded as linear functionals of the infinite dimensional control variable,

{A(t) : t ∈ [0, 1]}. Such programs are studied extensively in Anderson and Nash (1987), though we

will see momentarily that we need not solve this problem explicitly to characterize the equilibrium

distribution of applicant qualifications under a CB policy regime in this model.

Let µ0 be the multiplier on the capacity constraint and let θ be the multiplier on the represen-

tation constraint in the firm’s CB optimization problem stated above. Then, by the infinite dimen-

sional analogue of the Kuhn-Tucker Theorem, the solution to this problem is given by: {A∗(t) = 1,

t + θξ(π1, π2, t) ≥ µ0; and A∗(t) = 0, t + θξ(π1, π2, t) < µ0} Thus, an optimal acceptance policy

function has the property that, for almost every t, either A∗(t) = 1, or A∗(t) = 0. It is also clear

that, since r∗2 < r2, this optimal policy under color-blind affirmative action cannot be a threshold

policy. Hence, we may conclude that there will be levels of qualification t and s, with t < s, such

that A∗(t) = 1 and A∗(s) = 0.

Observe that any color-blind acceptance policy function, A(t), defines two numbers: Ae =

{Z

A(t)fe(t)dt}, e ∈ {0, 1}. Ae is the probability that an applicant who exerts effort e is accepted,

and ω[A1 − A0] is the expected benefit to an applicant from choosing action e = 1 instead of

e = 0. So, all acceptance policies that generate a common value for [A0−A1] must induce the same

behavioral response from applicants. Now, given that a firm’s representation target for group 2 is

r2, and given that its capacity implies c = λr1 + (1 − λ)r2, it follows that the firm’s constraints

under a CB policy regime can be written as follows:

πRA1 + (1− πR)A0 = rR, R = 1, 2.

From these equations it follows that (A1 −A0) =[c−r2]

[λ(π1−π2)] .

32

B. Applicant Behavior

Regardless of the regulatory regime, an applicant with exogenous characteristics (R, k) who

anticipates firms to employ the acceptance policy A(R, t) will behave in accordance with the effort

supply function, e∗(R, k), given as follows:

e∗(R, k) = 1 if ωZ[A(R, t)[f1(t)− f0(t)]dt] ≥ k, otherwise e∗(R, k) = 0.

That is, applicants choose to exert high effort if and only if their effort costs are below some

threshold. Therefore, anticipating the acceptance policy function, A(R, t), the proportion of group

R who end-up choosing the action e = 1 is given simply by:

π∗R = GR(ω

ZA(R, t)[f1(t)− f0(t)]dt), R = 1, 2

C. Equilibrium

An equilibrium in this model is an acceptance policy for firms, A∗(R, t), and an effort supply

function for applicants, e∗(R, k), that are mutual best responses to one another. We are now in

a position to describe equilibrium behavior by applicants and firms under the three regulatory

regimes. We require a bit more notation. Given a qualification threshold t and an acceptance

capacity c, such that F−10 (1 − c) ≤ t ≤ F−11 (1 − c), define bπ(t, c) as the solution to the equation:F (π, t) = 1− c. That is, bπ(t, c) is the proportion of an applicant population facing the qualificationthreshold, t, that would need to choose high effort if the acceptance rate in that population to just

equal c. Obviously, bπ(t, c) = [F0(t)− (1− c)]/[F0(t)− F1(t)]

Moreover, let ∆F (t) = [F0(t)− F1(t)], ∆G(k) = [G1(k)−G2(k)], and define the function: δ(r2) =

ω(c − r2)/λ. Finally, let π∗R be in fraction of group R who exert high effort equilibrium, and let

t∗R be the qualification threshold that applies to group R in equilibrium (relevant only when firms

employ a threshold policy.) It is straightforward to verify the following proposition.

Proposition 1 Equilibrium may be determined as follows:

(a) LF equilibrium is given by group-specific high effort rates (π∗1, π∗2) and an acceptance threshold

t∗ such that: bπ(t∗, c) = G(ω∆F (t∗)), and π∗R = GR(ω∆F (t∗)), R = 1, 2,

33

where firms’ acceptance policy is A∗(R, t) = 1, t ≥ t∗, and A∗(R, t) = 0, t < t∗, and applicants’

effort supply function is e∗(R, k) = 1, k ≤ ω∆F (t∗), and e∗(R, k) = 0, k > ω∆F (t∗).

(b) CS equilibrium is given by the pairs of group-specific high effort rates and acceptance thresh-

olds, (π∗1, t∗1) and (π

∗2, t

∗2), where:

bπ(t∗R, rR) = π∗R = GR(ω∆F (t∗R)), R = 1, 2.

Firms follow the CS threshold policy A∗(R, t) = 1, t ≥ t∗R, and A∗(R, t) = 0, t < t∗R, and applicants

follow the effort supply function e∗(R, k) = 1, k ≤ ω∆F (t∗R), and e∗(R, k) = 0, k > ω∆F (t∗R),

R = 1, 2.

(c) CB equilibrium entails firms choosing some feasible acceptance policy and applicants choos-

ing: e∗(R, k) = 1 if and only if k ≤ k∗, else e∗(R, k) = 0, where k∗ solves ∆G(k∗) = δ(r2)/k∗

Proof. The claims (a) and (b) are a transparent consequence of the definition of equilibrium.

Concerning (c), as noted above, capacity and representation constraints require that applicants

face the effort incentive:

ω(A1 −A0) = ω[c− r2]/[λ(π1 − π2)]

while optimal behavior by applicants implies that:

π1 − π2 = ∆G(ω(A1 −A0))

Combining these two equations to eliminate (A1 − A0), and identifying k∗ in (c) above with the

value in equilibrium of ω(A1 −A0) yields the stated result.

D. The Impact of Affirmative Action on Applicant Qualifications in Equilibrium

Let us now consider how the pursuit by firms of greater representation of the disadvantaged

affects the equilibrium distribution of the qualifications presented by applicants in the two groups.

We adopt the following notation: given firms’ acceptance capacity, c, let r∗2(c) denote the proportion

of group 2 applicants who are accepted in Laissez-faire equilibrium. Thus,

r∗2(c) = 1− F (G2[∆F (t∗)], t∗),where t∗ solves bπ(t∗, c) = G(ω∆F (t∗)).

We consider the impact of “weak” (that is, r2 “close to” r∗2(c)) and “strong” (that is, r2 “close

to” c) affirmative action goals. We wish to distinguish two cases under which the marginally

34

1π1

*

π 2*

π*

( )F c01 1− − ( )F c1

1 1− − 10 t*

( )$ ,π t c

~G1

G

~G2

t

π

π1*

π 2*

( )$ ,π t r1

( )$ ,π t c1−

( )$ ,π t r2

t*$t2$t10 1

π

t

~G1

G

~G2

(A) (B)

1π1

*

π 2*

π*

( )F c01 1− − ( )F c1

1 1− − 10 t*

( )$ ,π t c

~G1

G

~G2

t

π

π1*

π 2*

( )$ ,π t r1

( )$ ,π t c1−

( )$ ,π t r2

t*$t2$t10 1

π

t

~G1

G

~G2

1π1

*

π 2*

π*

( )F c01 1− − ( )F c1

1 1− − 10 t*

( )$ ,π t c

~G1

G

~G2

t

π

1π1

*

π 2*

π*

( )F c01 1− − ( )F c1

1 1− − 10 t*

( )$ ,π t c

~G1

G

~G2

t

π

π1*

π 2*

( )$ ,π t r1

( )$ ,π t c1−

( )$ ,π t r2

t*$t2$t10 1

π

t

~G1

G

~G2

π1*

π 2*

( )$ ,π t r1

( )$ ,π t c1−

( )$ ,π t r2

t*$t2$t10 1

π

t

~G1

G

~G2

(A) (B)

Figure 4: Laissez-faire (A) and Color-Sighted (B) Equilibrium

accepted applicant in the absence of any kind of affirmative action has either a low or a high level

of qualifications. Specifically, let t0 solve the equation: f1(t0)/f0(t0) = 1. If in LF equilibrium

t∗ < t0 we say that firms’ equilibrium acceptance standards are “loose” (because the marginally

accepted applicant has low qualifications), while if t∗ > t0 we say firms’ acceptance standards are

“tight”(because the marginally accepted applicant has high qualifications). Examination of Figure

4 should suffice to establish the following, which we state without proof.

Proposition 2 If standards are loose in Laissez—faire equilibrium, then the pursuit of sufficiently

weak CS affirmative action goals leads to an increase in qualifications among the advantaged group

and a decrease in qualifications among the disadvantaged group, widening the racial qualifications

gap relative to Laissez—faire. Moreover, tight standards in Laissez—faire equilibrium imply that weak

CS affirmative action decreases qualifications among the advantaged, and increases qualifications

among the disadvantaged, thereby narrowing the racial qualifications gap. When Laissez—faire equi-

librium standards are close to the margin between being loose or tight then sufficiently strong CS

affirmative action goals must decrease the qualifications of both groups.

35

( )∆G k

′k kmax k$k ( )k F t* *= ω∆

π π1 2* *−

$ $π π1 2−π π δ

1 2− =*

k

π π δ1 2− =

k

( )∆G k

′k kmax k$k ( )k F t* *= ω∆

π π1 2* *−

$ $π π1 2−π π δ

1 2− =*

k

π π δ1 2− =

k

Figure 5: Color-Blind Affirmative Action Under Elitism, δ∗ > δ

Finally, consider the effect of color-blind affirmative action on applicant qualifications in equi-

librium. To do so, we need one last definition. We will say that a selection problem is characterized

by “elitism” if there is no acceptance policy by firms that can induce “high cost” applicants to

exert high effort. (Here an applicant is said to have “high cost” if k > k0, where k0 solves the

equation: g1(k0)/g2(k0) = 1.) That is, elitism obtains when the structure of the situation is such

that ω∆F (t0) < k0. (Note that effort incentives are maximal at the acceptance threshold t = t0,

and that group disparity of qualifications is greatest when the marginal applicant to choose e∗ = 1

has cost just equal to k0). Figure 4, together with Proposition 1(c), can be used to establish the

following result:

Proposition 3 Suppose the condition of elitism obtains. Then for every affirmative action goal

r2, with r∗2(c) < r2 < c, there is a unique color-blind affirmative action equilibrium. Moreover, as

r2 rises, the level of qualifications in both groups declines, as does the qualifications gap between the

groups. In the limit, as r2 approaches c (population parity as a goal), the proportion of applicants

choosing high effort in the color-blind affirmative action equilibrium approaches zero. However,

36

there does not exist an equilibrium in this model under color-blind affirmative action that implements

the representation target of population proportionality, r2 = c.

In light of Figure 5, all of the claims in Proposition 3 are straightforward, except the last on

the impossibility of implementing an affirmative action goal of population parity in a color-blind

fashion. Here is a demonstration of this result: We know that, as the r2 approaches c, applicants’

incentive to exert high effort vanishes. This leads to a non-existence result when r2 = c because, if

all workers take effort e = 0, then the distributions of qualifications presented by workers in both

groups in equilibrium would have to be identical. But then, no representation constraint would

need to be imposed on firms to generate population parity. Absent such a constraint, firms would

want to behave as they do under Laissez Faire, accepting the fraction c of workers with the highest

qualifications. Yet, were firms to do this, lower cost workers in both groups would then have an

incentive to choose exert high effort, and the fact that group 2 is disadvantaged relative to group

1 would imply a failure to reach population parity in equilibrium. We conclude that when one

group is disadvantaged relative to the other there can exist no color-blind equilibrium achieving

population parity!

37

Asian Black Hispanic White Asian Black Hispanic White Asian Black Hispanic White Asian Black Hispanic WhiteCollege GPA Rank 48.07 23.66 24.60 53.16 39.61 27.52 34.86 53.70 47.88 21.15 38.41 55.10 49.65 16.52 27.29 53.71

(30.05) (22.03) (16.64) (27.57) (28.26) (25.64) (24.74) (27.74) (29.04) (19.75) (29.41) (26.88) (27.68) (16.72) (17.01) (27.73)SAT Math 640.44 486.54 521.11 638.53 623.52 514.58 547.14 596.41 654.30 500.61 503.33 633.18 716.85 562.73 595.22 692.03

(86.61) (93.76) (115.70) (78.06) (70.58) (78.13) (88.55) (69.97) (60.19) (73.55) (81.81) (65.74) (50.91) (77.59) (88.67) (60.55)SAT Verbal 569.56 485.96 500 616.32 546.14 522.50 507.62 592.59 585.70 491.02 510.37 607.19 671.67 552.42 593.48 664.29

(100.91) (104.75) (120.49) (81.81) (90.59) (95.20) (98.53) (71.35) (84) (72.03) (80.5) (70.46) (68.48) (74.12) (81.89) (72.09)HS Percentile 88.48 76.49 76.43 89.62 87.93 84.49 92.12 89.96 92.72 82.04 87.54 93.20 96.84 84.09 94.13 94.63

(8.5) (18.18) (14.60) (9.62) (9.64) (16.86) (6.35) (9.52) (7.65) (15.73) (9.63) (6.57) (3.35) (14.72) (5.29) (7.10)Mother College Edu .64 .46 .56 .80 .55 .46 .05 .58 .71 .51 .44 .81 .56 .48 .17 .78

(.48) (.51) (.51) (.40) (.50) (.51) (.22) (.49) (.46) (.51) (.51) (.39) (.50) (.51) (.38) (.42)Father College Edu .87 .41 .61 .86 .68 .46 .19 .66 .78 .43 .56 .81 .69 .55 .25 .86

(.33) (.50) (.50) (.34) (.47) (.51) (.40) (.47) (.42) (.50) (.51) (.39) (.47) (.51) (.44) (.35)Zip Income 44240 31792 33906 42507 43826 34686 37858 43082 50031 32308 35986 46573 47131 37812 37652 47712

(15686) (14492) (10871) (17081) (16953) (15744) (14689) (17050) (18541) (14401) (18637) (18233) (18477) (18257) (16561) (20198)Percent Asian in Zip .10 .02 .04 .03 .10 .02 .11 .03 .08 .03 .04 .03 .08 .04 .05 .03

(.16) (.03) (.03) (.04) (.14) (.01) (.09) (.05) (.10) (.04) (.03) (.05) (.09) (.05) (.05) (.04)Percent Black in Zip .05 .39 .14 .08 .04 .18 .03 .04 .04 .41 .05 .04 .06 .34 .12 .06

(.09) (.38) (.13) (.13) (.08) (.26) (.06) (.09) (.07) (.34) (.07) (.09) (.14) (.35) (.14) (.11)Percent Hisp in Zip .05 .05 .19 .05 .09 .09 .34 .04 .06 .09 .22 .04 .08 .08 .24 .04

(.06) (.09) (.10) (.09) (.13) (.14) (.32) (.07) (.10) (.14) (.31) (.07) (.12) (.12) (.21) (.06)Legacy .01 .02 .06 .05 .02 .04 0 .10 .07 .06 .11 .14 .06 0 0 .15

(.10) (.13) (.24) (.22) (.15) (.20) (0) (.30) (.26) (.24) (.32) (.35) (.23) (0) (0) (.35)Male .40 .47 .67 .46 n/a n/a n/a n/a n/a n/a n/a n/a .54 .45 .29 .55

(.49) (.50) (.49) (.50) n/a n/a n/a n/a n/a n/a n/a n/a (.50) (.51) (.46) (.50)N 91 59 18 646 88 24 21 471 130 49 27 374 54 33 23 408

All data are drawn from the College and Beyond Database, except for the demographic data on zipcodes which were attained from the 1990 Census. The numbers in parentheses are standard deviations. See the data appendix for further details of the construction of the variables.

College DCollege A College B College CTable 1: Descriptive Statistics of Students in Four Liberal Arts Colleges, by Race, 1989

Asian Black Hispanic White Asian Black Hispanic White Asian Black Hispanic WhiteCollege GPA Rank 59.45 25.44 28.84 52.17 51.92 20.16 41.70 53.06 55.92 33.02 47.87 51.03

(25.07) (20.68) (21.24) (28.36) (27.47) (20.51) (27.37) (27.69) (28.71) (27.09) (29.36) (28.25)SAT Math 731.18 612.44 641.60 706.54 690.47 541.78 623.57 651.41 614.43 515.05 569.25 605.48

(48.33) (65.31) (73.61) (59.41) (60.69) (81.80) (74.25) (73.17) (101.11) (73.55) (91.24) (80.83)SAT Verbal 648.49 595.24 598.6 650.69 578.22 523.24 550 583.63 549.65 502.2 508.88 557.17

(86.16) (65.78) (69.55) (69.51) (89.37) (82.59) (86.37) (70.13) (90.59) (99.62) (83.91) (80.85)HS Percentile 97.83 94.85 92.85 95.69 94.80 84.41 93.36 93.25 87.06 83.9 81.38 81.63

(3.16) (5.19) (8.41) (6.51) (8.05) (14.75) (10.65) (8.58) (17.45) (15.63) (12.31) (16.03)Mother College Edu .78 .55 .44 .73 .68 .47 .32 .65 .29 .20 .15 .20

(.41) (.50) (.50) (.44) (.47) (.50) (.48) (.48) (.46) (.41) (.36) (.40)Father College Edu .93 .49 .48 .84 .87 .45 .58 .79 .32 .17 .35 .24

(.25) (.50) (.50) (.37) (.33) (.50) (.50) (.41) (.47) (.38) (.48) (.43)Zip Income 51773 39858 41374 50741 45006 33223 33077 45470 35565 26042 35578 42848

(18964) (20783) (17922) (20460) (16146) (14568) (11168) (18715) (16959) (11383) (20796) (21054)Percent Asian in Zip .07 .03 .07 .04 .08 .03 .04 .03 .06 .02 .03 .03

(.12) (.03) (.11) (.04) (.13) (.04) (.05) (.05) (.08) (.04) (.03) (.05)Percent Black in Zip .05 .29 .03 .05 .05 .39 .11 .05 .16 .39 .11 .11

(.10) (.34) (.07) (.11) (.09) (.33) (.19) (.10) (.18) (.28) (.18) (.17)Percent Hisp in Zip .05 .06 .25 .04 .05 .06 .18 .03 .08 .04 .12 .05

(.07) (.07) (.29) (.07) (.06) (.11) (.23) (.06) (.13) (.08) (.18) (.08)Legacy .17 .05 .12 .22 .15 .06 .13 .25 .59 .72 .65 .72

(.38) (.22) (.33) (.41) (.35) (.21) (.34) (.43) (.50) (.45) (.48) (.45)Male .50 .48 .48 .58 .50 .41 .55 .52 .46 .52 .54 .52

(.50) (.50) (.50) (.49) (.50) .49) (.51) (.50) (.50) (.50) (.50) (.50)N 119 82 50 888 281 157 28 1353 68 98 48 1270All data are drawn from the College and Beyond Database, except for the demographic data on zipcodes which were attained from the 1990 Census. The numbers in parentheses are standard deviations. See the data appendix for further details of the construction of the variables.

College E College F College G

Table 2: Descriptive Statistics of Students in Three Research Universities, by Race, 1989

College A College B College C College D College E College F College GSAT Math 4.04(1.39) -.60 (1.78) 5.08 (1.69) 7.57(1.51) 6.33(0.89) 1.61(0.95) 9.41(1.86)

SAT Verbal 5.47(1.31) 8.98 (1.62) 7.15 (1.68) 12.85(1.30) 6.57(0.89) 5.77(0.95) 10.51(1.64)HS Percentile 3.12(1.11) 8.79 (1.42) 8.42 (1.84) 7.49(1.79) 6.06(0.54) 6.68(0.79) 8.37(1.69)

Mother College Edu 2.58(2.63) 8.40 (3.00) -3.45 (3.76) 3.61(2.15) 7.47(3.40) 3.00(1.46) 3.54(2.65)Father College Edu 4.35(2.99) -3.76 (3.60) 6.07 (4.04) 5.48(2.84) 5.52(4.06) 4.34(1.72) 3.46(3.12)

Zip Income -0.04(0.64) -1.44 (.80) -.47 (0.72) -0.74(0.44) 0.33(0.40) 0.87(0.38) 0.03(0.61)Legacy 4.66(4.55) .59 (4.05) .65 (3.64) -0.47(1.96) -0.39(2.77) -3.23(1.54) 3.46(3.43)

Pct. Asian in Zip 14.07(16.83) 16.78 (16.82) 6.28 (19.45) 33.05(13.58) 21.89(13.47) 9.74 (9.39) -8.53(23.94)Pct. Black in Zip -11.72(5.78) -29.10 (10.99) -14.26 (7.59) -15.91(5.31) -9.29(4.03) -21.86(4.03) -13.92(7.33)Pct. Hisp in Zip -15.76(11.21) -22.15 (11.51) -.42 (11.40) -3.24(9.10) 0.83(7.84) -13.78 (9.01) -26.31(13.65)

Male -4.77(2.06) n/a n/a -7.92(1.66) -10.07(1.53) -7.17 (1.34) -2.16(2.20)

R-Squared 0.16 0.21 0.19 0.37 0.28 0.16 0.26Number of Obs. 761 429 512 494 955 1787 1419

Notes: College rank is percentiles in distribution of cumulative GPA among students who matriculated at that college in 1989. HS Percentile is students'

percentile in his high school. Mother and Father’s Education are dummies for students’ mother and father being college educated. Zip Income is the median

income of the student’s zip code from the 1990 Census. Bold coefficients are five percent significant. Increments: SAT variables 100 points,

HS Percentile 10 percentiles, ZIP Income 10000 dollars. We used dummies for the missing data. (Coefficients for these variables are not reported in this table).

Table 3: Performance Equation: Predicted College Rank

College A College B College C College D College E College F College GSAT Math -0.06(0.01) -.06 (.01) -.07 (.01) -0.10(0.02) -0.10(0.01) -0.08 (0.01) -0.06(0.01)SAT Verbal -0.02(0.01) -.03 (.01) -.06 (.01) -0.04(0.01) -0.03(0.01) -0.02 (0.01) -0.03(0.01)

HS Percentile -0.04(0.01) .01 (.01) -.07 (.01) -0.05(0.01) 0.01(0.01) -0.04 (0.01) 0.01(0.00)

Mother College Edu 0.02(0.02) .01 (.02) .03 (.03) 0.01(0.02) 0.04(0.02) -0.00 (0.01) -0.02(0.03)Father College Edu -0.11(0.02) -.04 (.02) -.08 (.03) -0.02(0.03) -0.09(0.02) -0.05 (0.01) 0.02(0.04)

Zip Income 0.01(0.00) .01 (.01) .00 (.01) 0.01(0.01) 0.01(0.00) 0.00 (0.00) 0.00(0.00)Legacy 0.01(0.03) -.02 (.03) -.01 (.03) -0.09(0.03) -0.06(0.02) -0.05(0.01) -0.04(0.02)Pct. Asian in Zip -0.25(0.12) -.06 (.11) -.10 (.15) 0.14(0.20) 0.00(0.11) -0.01 (0.08) 0.11(0.12)Pct. Black in Zip 0.57(0.04) .57 (.07) .85 (.06) 0.57(0.06) 0.64(0.04) 0.78 (0.03) 0.44(0.04)Pct. Hisp in Zip -0.02(0.08) -.02 (.08) -.08 (.09) -0.13(0.12) -0.15(0.08) -0.00(0.07) 0.05(0.07)Male 0.01(0.02) n/a n/a 0.01(0.02) 0.02(0.01) 0.01(0.01) 0.04(0.01)

R-Squared .41 .22 .55 .37 .39 .42 .21Number of Obs. 761 429 512 494 955 1787 1419

Notes: Dependent variable is student's probability of being black. HS Percentile is students' percentile in his high school. Mother and Father’s Education are dummies for students mother and father being College Educated. Zip Income is the averageincome of the student’s zip code from the 1990 Census. Increments: SAT variables 100 points, HS Percentile 10 percentiles, ZIP Income 10000 dollars. We used dummies for the missing data. (Coefficients for these variables are not reported in this table).

Table 4: Race Equation: Probability of Being Black

College A College B College C College D College E College F College G Average

Random Admissions 86.29 87.22 80.43 85.04 80.67 85.62 81.26 83.62

Laissez-Faire W/O SAT 96.96 98.50 97.00 93.90 95.01 98.45 96.65 96.88

Laissez-Faire W/O 99.77 98.02 97.89 99.58 99.56 98.67 97.24 98.57HS Percentile

Color-Sighted 97.97 99.55 97.64 97.43 98.66 98.59 99.77 98.68

Color-Blind 94.28 98.67 95.33 90.82 96.40 95.95 98.74 96.16Predicted College Rank of a student is estimated by the OLS regression. For each policy, we compute the average predicted College Rank of the admitted class. We call this value the Performance of the Policy. Tocompute the Relative Performance, we index Laissez Faire's Performance as 100. For example color-sighted RelativePerformance=(color-sighted Performance *100)/(Laissez-Faire Performance). Average is the population weighted average.

Table 5: Relative Performances of Color-Blind and Color-Sighted Policies, by Race Constraint

College A College B College C College D College E College F College GLaissez-Faire Asians 34 30 62 26 74 174 41 Blacks 2 5 1 0 4 11 26 Hispanics 2 3 5 1 7 10 14 Whites 376 265 223 232 484 734 662

Color-Sighted Asians 29 30 53 24 73 165 38 Blacks 29 12 24 16 41 78 49 Hispanics 2 3 5 1 5 9 16 Whites 354 257 208 217 449 676 639

Color-Blind Asians 32 28 52 25 71 165 43 Blacks 27 12 26 17 43 73 51 Hispanics 5 4 7 4 8 10 16 Whites 349 258 205 213 447 680 633

This table shows the effects of Laissez-Faire, Color-Sighted and Color-Blind policies on the composition of the admitted class. Notethat Color-Blind policy is targeted for black students only. The number of students are not integers under Color-Blind policy. This isbecause, the college assigns an admission probability for two students strictly between zero and one. The college has a linear programmingproblem: to maximize a linear function with respect to linear constraints. The college assigns a value to each student (with respect tohis expected college rank and probability of being black) and admits the students with higher values. However, on the margin the college isindifferent between two students. (two students have equal values). That is why admitting a fraction of these students is consistent with theresult being optimal.

Table 6: Diversity of Entering Class Under Alternative Policies

LF CB LF CB LF CB LF CB LF CB LF CB LF CB LF CB LF CBCollege A 4.04 0.16 5.47 4.18 3.12 0.53 2.58 3.87 4.35 -2.76 -0.04 0.61 -11.72 25.14 14.07 -2.10 -15.76 -17.05College B -0.60 -4.88 8.98 6.84 8.79 9.50 8.40 9.11 -3.76 -6.61 -1.44 -0.73 -29.10 11.56 16.78 12.50 -22.15 -23.58College C 5.08 1.68 7.15 4.24 8.42 5.02 -3.45 -1.99 6.07 2.18 -0.47 -0.47 -14.26 27.02 6.28 1.42 -0.42 -4.31College D 7.57 -3.74 12.85 8.33 7.49 1.83 3.61 4.74 5.48 3.22 -0.74 0.39 -15.91 11.77 33.05 48.89 -3.24 -17.94College E 6.33 -0.47 6.57 4.53 6.06 6.74 7.47 10.19 5.52 -0.60 0.33 1.01 -9.29 34.20 21.89 21.89 0.83 -9.36College F 1.61 -1.96 5.77 4.88 6.68 4.89 3.00 3.00 4.34 2.11 0.87 0.87 -21.86 12.98 9.74 9.29 -13.78 -13.78College G 9.41 6.38 10.51 9.00 8.37 8.87 3.54 2.53 3.46 4.47 0.03 0.03 -13.92 8.30 -8.53 -2.98 -26.31 -23.79

Pct. Hispanic

Table 7: Weight on Students' Characteristics in the Admission Formula for Laissez-Faire and Color-Blind Policies, by Representation Goal

Father EDU IncomeSAT Math SAT Verbal HS Pct. Mother EDU Pct. AsianPct. Black